On (omega)-(beta) Continuous Functions

Abstract

A subset $A$ of a topological space $X$ is said to be $\omega \beta -$open  if for every $x\in A$ there exists a $\beta -$open set $U$ containing $x$ such that $U-A$ is a countable. In this paper, we introduce and study a new class of functions  called $\omega \beta -$continuous functions by using the notion of $\omega \beta -$open sets. In particular we say a function $f:X\to Y$  is $\omega \beta -$continuous  if and only if for each $x\in X$ and each open set $V$ in $Y$ containing $f(x)$ there exists an $\omega \beta -$open set $U$ containing $x$ such that $f(U)\subseteq V$. We give some characterizations of $\omega \beta -$continuous functions, introduce and study $\omega \beta -$irresolute and $\omega \beta -$open functions. Finally, we investigate the relationship between these type of functions.